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^{10}C_1+^{10}C_3+^{10}C_5+^{10}C_7+^{10}C_9=

Question

$^{10} C_{1} +^{10} C_{3} +^{10} C_{5} +^{10} C_{7} +^{10} C_{9} =$

2^{9}

2^{10}

2^{10} - 1

None of these

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^{10} C_{1} +^{10} C_{3} +^{10} C_{5} +^{10} C_{7} +^{10} C_{9} =

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Text solutionVerified

For

(X + A)^{n} = C_{0}^{n} X^{n} + \dots C_{r}^{n} X^{n - r} A^{r} + \dots C_{n}^{n} A^{n}

Put

X = 1; A = - 1

Hence,

0 = C_{0}^{n} - C_{1}^{n} + C_{2}^{n} - C_{3}^{n} \dots

and so on

\Rightarrow C_{0}^{n} + C_{2}^{n} + C_{4}^{n} + \dots = C_{1}^{n} + C_{3}^{n} + C_{5}^{n} + \dots

...(1)

Now put

A = 1

and we see that

2^{n} = C_{0}^{n} + C_{1}^{n} + C_{2}^{n} + C_{3}^{n} + C_{4}^{n} + C_{5}^{n} + \dots

...(2)

Hence, the sum of

C_{1}^{n} + C_{3}^{n} + C_{5}^{n} + \dots = 2^{n - 1}

(From (1) and (2))

In the problem,

n = 10

Hence, answer is

2^{9}

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